CH3M.DOC 01/14/2019
CHAPTER OBJECTIVES - CHAPTER 3
The Interest Factor in Financing
The student who has successfully completed this chapter should be able to perform the following:
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Calculate the present value of a single sum to be received in the future given a required rate of interest (the discount rate) and the time to maturity.
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Calculate the future value of a single sum compounded at some rate of interest over a specified time period.
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Define the term annuity and calculate the present value or future value of an annuity.
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Calculate the effective annual yield for a series of cash flows assuming annual, semiannual, quarterly, monthly, and daily compounding.
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Define what is meant by the internal rate of return on an investment (IRR) and calculate it for a series of cash flows.
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Using the effective annual yield, calculate the equivalent nominal annual rate of an investment.
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Solutions to Questions - Chapter 3
The Interest Factor in Financing
Question 3-1
What is the essential concept in understanding compound interest?
The concept of earning interest on interest is the essential idea that must be understood in the compounding process and is the cornerstone of all financial tables and concepts in the mathematics of finance.
Question 3-2
How are the interest factors Exhibit 3-3 developed?
Computed from the general formula for compounding for monthly compounding for various combinations of “i” and years. FV = PV x (1+I)n.
Question 3-3
What general rule can be developed concerning maximum values and compounding intervals within a year? What is an equivalent annual yield?
Whenever the nominal annual interest rates offered on two investments are equal, the investment with the more frequent compounding interval within the year will always result in a higher effective annual yield. An equivalent annual yield is a single, annualized discount rate that captures the effects of compounding (and if applicable, interest rate changes).
Question 3-4
What does the time value of money mean? How is it related to present value? What process is used to find present value?
Time value simply means that if an investor is offered the choice between receiving $1 today or receiving $1 in the future, the proper choice will always be to receive the $1 today, because that $1 can be invested in some opportunity that will earn interest. Present value introduces the problem of knowing the future cash receipts for an investment and trying to determine how much should be paid for the investment at present. When determining how much should be paid today for an investment that is expected to produce income in the future, we must apply an adjustment called discounting to income received in the future to reflect the time value of money.
Question 3-5
How does discounting, as used in determining present value, relate to compounding, as used in determining future value? How would present value ever be used?
The discounting process is a process that is the opposite of compounding. To find the present value of any investment is simply to compound in a “reverse” sense. This is done by taking the reciprocal of the interest factor for the compound value of $1 at the interest rate, multiplying it by the future value of the investment to find its present value.
Present value is used to find how much should be paid for a particular investment with a certain future value at a given interest rate.
Question 3-6
What are the interest factors in Exhibit 3-9? How are they developed?
Compound interest factors for the accumulation of $1 per period, e.g., $1 x [1 + (1+i) + (1+i)2 …] etc.
Question 3-7
What is an annuity? How is it defined? What is the difference between an ordinary annuity and an annuity due?
An annuity is a series of equal deposits or payments.
An ordinary annuity assumes payments or receipts occur at the end of a period.
An annuity due assumes deposits or payments are made at the beginning of the period.
Question 3-8
Why can’t interest factors for annuities be used when evaluating the present value of an uneven series of receipts? What factors must be used to discount a series of uneven receipts?
With an annuity, interest factors can be summed because the payments are equal in amount and are received at equal intervals. If the series of annuities being evaluated is uneven, the interest factors cannot be summed and the interest factors for annuities are of value mathematically. In the case of an uneven series of receipts, the calculation requires the use of ordinary present value factors.
Question 3-9
What is the sinking-fund factor? How and why is it used?
A sinking-fund factor is the reciprocal of interest factors for compounding annuities. These factors are used to determine the amount of each payment in a series needed to accumulate a specified sum at a given time. To this end, the specified sum is multiplied by the sinking-fund factor.
Question 3-10
What is an internal rate of return? How is it used? How does it relate to the concept of compound interest?
The internal rate of return integrates the concepts of compounding and present value. It represents a way of measuring a return on investment over the entire investment period, expressed as a compound rate of interest. It tells the investor what compound interest rate the return on an investment being considered is equivalent to.
Solutions to Problems - Chapter 3
The Interest Factor in Financing
Problem 3-1
a)Future Value=$12,000 (FVIF, 9%, 7 years)
=$12,000 (1.82804)
=$21,936 (annual compounding)
b)Future Value=$12,000 (QFVIF, 9% , 7 years)
=$12,000 (1.86454)
=$22,375 (quarterly compounding)
c) Equivalent annual yield: (consider one year only)
Future Value of (a)=$12,000 (FVIF, 9%, 1 year)
=$12,000 (1.09)
=$13,080
($13,080 - $12,000) / $12,000 = 9.00% effective annual yield
Future Value of (b)=$12,000 (QFVIF, 9%, 1 year)
=$12,000 (1.09308)
=$13,117
($13,117 - $12,000) / $12,000 = 9.31% effective annual yield
Alternative (b) is better because of its higher effective annual yield.
Problem 3-2
Investment A: 7% compounded monthly
Future Value of A=$25,000 (MFVIF, 7%, 1 year)
=$25,000 (1.07229)
=$26,807 (monthly compounding)
Investment B: 8% compounded annually
Future Value of B=$25,000 (MFVIF, 8%, 1 year)
=$25,000 (1.08)
=$27,000 (annual compounding)
Investment B should be chosen over A. Investment B that pays 8% compounded annually is the better choice because it provides the greater future value and therefore the greater effective annual yield.
Problem 3-3
Find the present value of $15,000 discounted at an annual rate of 10% for 10 years.
Present Value=$15,000 (PVIF, 10%, 10 years)
=$15,000 (.38554)
=$5,783 (annual compounding)
The investor should not purchase the lot because the present value of the lot (discounted at the appropriate interest rate) is less than the current asking price of $7,000.
If the investor purchases the lot, the effective yield on the investment will be ( ^ indicated exponent):
Effective Yield=(15,000/7,000) ^ .1 - 1
=(2.14286) ^ .1 - 1
=7.9% (rounded)
Problem 3-4
Find the present value of $45,000 discounted at an 18% annual rate compounded quarterly for a six year period.
Present Value=$45,000 (QPVIF, 18%, 6 years)
=$45,000 (.34770)
=$15,647 should be paid today
Note that a quarterly interest factor is used in this problem because the investor indicates that an annual rate of 18% is desired.
Problem 3-5
Find the future value of 24 end-of-period payments of $5,000 at an annual rate of 8.5%, compounded semi-annually.
Future Value=$5,000 (SAFVIFA, 8.5%, 12 years) - ordinary annuity
=$5,000 (40.36113)
= $201,806
Note: Total cash deposits are $5,000 x 24 = $120,000. Total interest equals $81,806 ($201,806 - $120,000). These semi-annual deposits constitute an annuity. The $120,000 represents the return of capital of initial principal while the $81,806 represent the interest earned on the capital contributions.
Find the future value of 24 beginning-of-period payments of $5,000 at an annual rate of 8.5%, compounded semi-annually.
Future Value=$5,000 (SAFVIFA, 8.5%, 12 years) - annuity due
=$5,000 (42.07648)
= $210,382
Problem 3-6
Find the future value of 4 years of quarterly payments at $1,250 each earning an interest rate of 15 percent annually, compounded quarterly.
Future Value=$1,250 (QFVIFA, 15%, 4 years)
=$1,250 (21.39274)
= $26,741
Problem 3-7
Year / Amount Deposited / FVIF / Future Value1 / $2,500 / x (FVIF, 9%, 4 yrs.) or / 1.41158 / $3,529
2 / $0 / x (FVIF, 9%, 3 yrs.) or / 1.29503 / $0
3 / $750 / x (FVIF, 9%, 2 yrs.) or / 1.18810 / $891
4 / $1,300 / x (FVIF, 9%, 1 yr.) or / 1.09000 / $1,417
5 / $0 / $0
Total Future Value = $5,837
The investor will have $5,837 on deposit at the end of the 5th year.
*Each deposit is made at the end of the year.
Problem 3-8
Find the present value of 10 end-of-year payments of $2,150 discounted at an annual interest rate of 18 percent.
Present Value=$2,150 (PVIFA, 18%, 10 years) - ordinary annuity
=$2,150 (4.49409)
=$9,662 should be paid today
Find the present value of 10 beginning-of-year payments of $2,150 discounted at an annual interest rate of 18 percent.
Present Value=$2,150 (PVIFA, 18%, 10 years) - annuity due
=$2,150 (5.30302)
= $11,401 should be paid today
Problem 3-9
a) Find the present value of 8 years of monthly payments, or 96 payments, of $750 (end-of-month) discounted at an interest rate of 17 percent compounded monthly.
Present Value=$750 (MPVIFA, 17%, 8 years) - ordinary annuity
=$750 (52.29728)
= $39,223 should be paid today
b) The total sum of cash received over the next 8 years will be:
8 years x 12 payments per year x $750 per month = $72,000
c)Total cash received by the investor$72,000
Initial price paid by the investor$39,223
Difference: Interest Earned$32,777
The difference represents the total interest earned by the investor on the initial investment of $39,223 if each $750 payment received earns 17 percent compounded monthly.
Problem 3-10
Year / Amount Deposited / MPVIF / Present Value1 / $12,500 / x (MPVIF, .75%, 12 months) or / 0.91424 / $11,428
2 / $10,000 / x (MPVIF, .75%, 24 months) or / 0.83583 / $8,358
3 / $7,500 / x (MPVIF, .75%, 36 months) or / 0.76415 / $5,731
4 / $5,000 / x (MPVIF, .75%, 48 months) or / 0.69861 / $3,493
5 / $2,500 / x (MPVIF, .75%, 60 months) or / 0.63870 / $1,597
6 / $0 / x (MPVIF, .75%, 72 months) or / 0.58392 / $0
7 / $12,500 / x (MPVIF, .75%, 84 months) or / 0.53385 / $6,673
Total Present Value = $37, 280
* Each deposit is made at the end of the year
The investor should pay no more than $37,280 for the investment in order to earn the 9% annual interest rate compounded monthly.
Problem 3-11
Annual sinking fund payments required to accumulate $50,000 after ten years
Future Value=Payment x (FVIFA, 10%, 10 years)
Payment=Future Value / (FVIFA, 10%, 10 years)
=$50,000 / (15.93742)
=$3,137 per year
Note to Instructor: In problem 3-11(b), the text requests that annual payments be calculated, that is,12 x monthly payments.
Monthly sinking fund payments required to accumulate $50,000 after ten years.
Future Value=Payment x (FVIFA, 10%, 10 years)
Payment=Future Value / FVIFA, 10%, 10 years)
=$50,000 / (204.84498)
=$244.09 per month
Note: Some text use a sinking fund factor (SFF) that can be multiplied by the desired ending amount in order to find the payment.
As an example, $50,000 x (SFF, 10%, 10 years) would give the $3,137 annual payment. The SFF is nothing more than 1/ (FVIFA, 10%, 10 years) or the reciprocal of the appropriate future value of an annuity interest factor.
Problem 3-12
What will be the rate of return (yield) on a project that initially costs $100,000 and is expected to pay out $15,000 per year for the next ten years?
The solution must be interpolated:
Present Value=Payment x (PVIFA, ?%, 10 years)
$100,000=$15,000 (PVIFA, ?%, 10 years)
6.66667=(PVIFA, ?%, 10 years)
Looking to the annual tables, we are seeking a value of 6.66667 in the column for a 7 year period. This value falls between 7% and 10%. Interpolation yields the following estimate:
PVIFA @ 7%=7.023582PVIFA @ 7%=7.023582
PVIFA @ 10%=6.144567Desired PVIFA=6.666667
0.8790140.356915
0.356915 / 0.879014 x (10.0% - 7.0%)=1.22%
7.0% + 1.22%=8.22%
An exact calculator solution is:8.14%
This is a poor investment for Buildsmart because the IRR of 8.14% does not exceed Buildsmart’s desired return of 9%.
Problem 3-13
What will be the rate of return (yield) on a project that initially costs $75,000 and is expected to pay out $1,020 per month for the next 25 years?
The solution must be interpolated:
Present Value=Payment x (MPVIFA, ?%, 300 months)
$75,000=$1,020 (MPVIFA, ?%, 300 months)
73.52941=(MPVIFA, ?%, 300 months)
In the annuity tables, we are seeking a value of 73.529411765 in the MPVIFA column for a 25 year period. This value falls between 15% and 20%. Interpolation yields the following estimate:
MPVIFA @ 15%=78.07434MPVIFA @ 15%=78.07434
MPVIFA @ 20%=59.57872Desired PVIFA=73.52941
18.495624.54493
4.544925 / 18.49562 x (20.0% - 15.0%)=1.23%
15.0% + 1.23%=16.23%
An exact calculator solution is:16.01%
The total cash received will be: $1,020 x 25 years x 12 months = $306,000
How much is profit and how much is return on capital?
Total Amount Received$306,000
Total Capital Invested (returned)$75,000
Total Profit (interest earned)$231,000
The total cost of the investment, $75,000, is capital recovery.
The difference between the total amount received and the capital recovery is total profit earned.
Problem 3-14
(a)
Year / Amount Received* / MPVIF / Present Value1 / $5,500 / x (PVIF, 13%, 1 year) or / 0.88496 / $4,867
2 / $7,500 / x (PVIF, 13%, 2 year) or / 0.78315 / $5,874
3 / $9,500 / x (PVIF, 13%, 3 year) or / 0.69305 / $6,584
4 / $12,500 / x (PVIF, 13%, 4 year) or / 0.61332 / $7,666
Total Present Value = $24,991
The investor should pay not more than $24,991 for investment in order to earn the 13 percent annual interest rate compounded annually.
(b)
Year / Amount Received* / MPVIF / Present Value1 / $5,500 / x (MPVIF, 13%, 1 year) or / 0.87871 / $4,833
2 / $7,500 / x (MPVIF, 13%, 2 year) or / 0.77213 / $5,791
3 / $9,500 / x (MPVIF, 13%, 3 year) or / 0.67848 / $6,446
4 / $12,500 / x (MPVIF, 13%, 4 year) or / 0.59619 / $7,452
Total Present Value = $24,522
The investor should pay not more than $24,533 for investment in order to earn the 13 percent annual interest rate compounded monthly.
(c) These two amounts are different because the return demanded in part (b) is compounded monthly.
Problem 3-15
What will be the rate of return (yield) on a project that initially costs $100,000 and is expected to receive $1,500 per month for the next 5 years and, at the end of the five years, return the initial investment of $100,000?
Manually, the solution must be determined through a trial and error process of solving the following question:
Present Value=Payment (MPVIFA, ?%, 60 months) + Principal (PVIF, ?%, 60 months)
$100,000=$1,500 (MPVIFA, ?%, 60 months) + $100,000 (PVIF, ?%, 60 months)
The internal rate of return for the above investment is 18%.
Problem 3-16
a) Find the ENAR for 10% EAY - Monthly Compounding.
ENAR=[( 1 + EAY) ^ (1/m) - 1] x m
=[( 1 + .10) ^ (1/12) -1] x 12
=[ 1.00797414 - 1] x 12
=[.00797414] x 12
=.09568968 or 9.57%
b) Find the ENAR for 10% EAY - Quarterly Compounding
ENAR=[( 1 + EAY) ^ (1/m) x m
=[( 1 + .10) ^ (1/4) -1] x 4
=[ 1.0241137 - 1] x 4
=[.0241137] x 4
=.0964548 or 9.56%
Problem 3-17
Part 1, calculate annual returns compounded annually:
(Note: calculator should be set for one payment per period)
The Annual Rate compounded Monthly:
Solution:
N=28
PMT=1,000
PV=-24,000
FV=0
Solve for the yield:
I=1.09577
The monthly rate can now be used to calculate the equivalent annual rate.
The Annual Rate compounded annually = 1.139714
Solution:
PV=-1
I=1.09577
PMT=0
N=12
Solve for the future value:
FV=1.139714
The annual rate of interest (compounded annually) needed to provide a return equivalent to that of an annual rate compounded monthly is:
FV - PV = 1.139714 - 1.0 =13.9714%
This return is far greater than the annual rate compounded monthly
Mo.yld x no. mos.=1.0957x12 = 13.1492%
This tells us that an investor would have to find an investment yielding 13.9714% if compounding occurred on an annual basis (once per year) in order for it to be equivalent to an investment that provides an annual rate of 13.1492% compounded monthly.
Chapter 3: The Interest Factor in Financing
Sample Exam Questions
The following questions are multiple choice.
MC 3-1
The future value of a single deposit of $1,000 will be greater when this amount is compounded:
A.annually
B.semi-annually
C.quarterly
D.monthly
MC 3-2
The future value of $1,000 compounded annually for 8 years at 12% may be calculated with the following formula:
FV = $1,000 * (1 + 12%)8
If the same $1,000 was compounded quarterly, what formula would you use to calculate the FV?
A.FV = $1,000 * (1 + 3%)8
B.FV = $1,000 * (1 + 12%)32
C.FV = $1,000 * (1 + 3%)32
D.FV = $1,000 * (1 + 12%)2
MC 3-3
If you saw a table containing the following factors, what kind of interest factor would you be looking at?
End of Year / 6%1 / 1.06000
2 / 1.12360
3 / 1.19101
4 / 1.26247
5 / 1.33822
A.Present value of a single amount
B.Future value of a single amount
C.Present value of an annuity
D.Future value of an annuity
MC 3-4
Begin with a single sum of money at period 0. First, calculate a future value of that sum at 12.01%. Then discount that future value back to period 0 at 11.99%. In relation to the initial single sum, the discounted future value:
A.is greater than the original amount.
B.is less than the original amount.
C.is the same as the original amount.
D.cannot be determined with the information given.
MC 3-5
The future value compound factor given for period (n) at 15%:
A.would be less than the factor for period (n+1) at 15%.
B.would be greater than the factor given for period (n+1) at 15%.
C.would be the same as the factor given for period (n+1) at 15%.
D.bears no relationship to the factor for period (n+1) at 15%.
MC 3-6
Which of the following is not a basic component of any compounding problem?
A.An initial deposit
B.An interest rate
C.A period of time
D.A net present value
MC 3-7
If an investment earns 12% annually,
A.an equivalent monthly investment would have to earn a higher equivalent nominal rate to yield the same return.
B.an equivalent monthly investment would have to earn a lower equivalent nominal rate to yield the same return.
C.an equivalent monthly investment would have to earn the same equivalent nominal rate to yield the same return.
D.a relation cannot be determined between a monthly and annual investment.
MC 3-8
The internal rate of return:
A.is also known as the investment of investor’s yield.
B.represents a return on investment expressed as a compound rate of interest.
C.is calculated by setting the price of an investment equal to the stream of cash flows it generates and solve for the interest rate.
D.can be defined by all of the above.
For questions 3-9 and 3-10.
Present Value Factor for Reversion of $1
Period / 6% / 7% / 8% / 9% / 10%1 / .943396 / .934579 / .925926 / .917431 / .909091
2 / .889996 / .873439 / .857339 / .841680 / .826446
3 / .839619 / .816298 / .793832 / .772183 / .751315
4 / .792094 / .762895 / .713503 / .708425 / .683013
5 / .747258 / .712986 / .680583 / .644931 / .620921
6 / .704961 / .666643 / .630170 / .596267 / .564474
MC 3-9
Using only the information above, what would the IRR be for an investment that cost $500 in period 0 and was sold for $750 in period 5?
A.Between 6% and 7%
B.Between 7% and 8%
C.Between 8% and 9%
D.Between 9% and 10%
MC 3-10
Using only the information above, approximately how much would you pay today for an investment that pays $0 annual interest, but earns 8% interest over the next four years and has a face value at maturity of $13,500?
A.$8,000
B.$9,000
C.$10,000
D.$11,000
The following questions are true-false.
TF 3-11In order to solve a compounding problem, you must know all four of its basic parts.
TF 3-12One way to calculate the present value of a single payment is with the following formula: PV = FV * (1+i)n.
TF 3-13At 6%, the present value of a $1 payment in 12 months is .941905. At 7%, the present value of a $1 payment in 12 months is .950342.
TF 3-14The future value of $800 deposited today would be greater if that deposit earned 8% rather than 7.75%.
TF 3-15You are able to calculate the present value of an annuity by understanding the following relationship:
FV = PV (1+I)
TF 3-16You usually see an ordinary annuity used in business and rarely see an annuity due used in business.
TF 3-17The internal rate of return is the good feeling you get inside when you earn a return on your investment.
TF 3-18An investment may have more than one internal rate of return.
TF 3-19Assume that an investment, with an single initial cost of $1,000 and a yield of $50 monthly for 10 years, had a 7% IRR in the 60th month and a 7.2% IRR five months later. The IRR can be 6.8% in the 62nd month.
TF 3-20The future value of a $1 annuity compounded at 5% annually is greater than the future value of a $1 annuity compounded at 5% semi-annually.
SOLUTIONS TO CHAPTER 3
Multiple Choice
3-1.D
3-2.C
3-3.B
3-4.A
3-5.A
3-6.D
3-7.B
3-8.D
3-9.C
3-10.C
True/False
3-11.F
3-12.F
3-13.F
3-14.T
3-15.T
3-16.F
3-17.F
3-18.T
3-19.F
3-20.F
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